On decreasing each side of an equilateral triangle by 2 cm, there is a decrease of 4 `sqrt3` `cm^2` in its area. The length of each side of the triangle is
एक समभुज त्रिभुज की प्रत्येक भुजा को 2cm कम करने से उसके क्षेत्रफल में 4`sqrt3 cm^2` की कमी हो जाती है। त्रिभुज की प्रत्येक भुजा की लंबाई कितनी है?

Options

A. 8 cm
B. 3 cm
C. 5 cm
D. 6 cm

Explanation

To find the length of each side of the equilateral triangle, we can follow these steps:

### Step 1: Let the length of each side of the equilateral triangle be \( L \) cm.

### Step 2: Calculate the area of the original triangle.
The area \( A \) of an equilateral triangle with side length \( L \) is given by the formula:
\[ A = \frac{\sqrt{3}}{4} L^2 \]

### Step 3: Determine the new side length after decreasing each side by 2 cm.
The new side length after decreasing each side by 2 cm is:
\[ L - 2 \text{ cm} \]

### Step 4: Calculate the area of the new triangle.
The area \( A' \) of the new triangle with side length \( L - 2 \) is:
\[ A' = \frac{\sqrt{3}}{4} (L - 2)^2 \]

### Step 5: Set up the equation based on the decrease in area.
According to the problem, the decrease in area is \( 4\sqrt{3} \) cm². Therefore, we can write the equation:
\[ A - A' = 4\sqrt{3} \]
Substituting the areas from Steps 2 and 4:
\[ \frac{\sqrt{3}}{4} L^2 - \frac{\sqrt{3}}{4} (L - 2)^2 = 4\sqrt{3} \]

### Step 6: Simplify the equation.
Factor out \( \frac{\sqrt{3}}{4} \):
\[ \frac{\sqrt{3}}{4} \left( L^2 - (L - 2)^2 \right) = 4\sqrt{3} \]
Now, simplify \( (L - 2)^2 \):
\[ (L - 2)^2 = L^2 - 4L + 4 \]
Thus, the equation becomes:
\[ \frac{\sqrt{3}}{4} \left( L^2 - (L^2 - 4L + 4) \right) = 4\sqrt{3} \]
This simplifies to:
\[ \frac{\sqrt{3}}{4} (4L - 4) = 4\sqrt{3} \]

### Step 7: Eliminate the common factor.
Multiply both sides by \( 4 \):
\[ \sqrt{3} (4L - 4) = 16\sqrt{3} \]
Now, divide both sides by \( \sqrt{3} \):
\[ 4L - 4 = 16 \]

### Step 8: Solve for \( L \).
Add 4 to both sides:
\[ 4L = 20 \]
Now divide by 4:
\[ L = 5 \text{ cm} \]

### Conclusion
The length of each side of the triangle is \( 5 \) cm.

---